Paperfolding, Automata, and Rational Functions - Diagonals and ...

Breaking Up in Characteristic p
The following breaking up procedure is fundamental below:
{x α = x α 1
1
· · · x αn
n : α ∈ S} is a basis for Fp[[x]] over ` Fp[[x]] ´ p .
Hence if y(x) ∈ Fp[[x]] and S = {0, 1, . . . , p − 1} n then, for α ∈ S ,
there are unique yα(x) ∈ Fp[[x]] such that y(x) = P
α∈S x α y p α(x) .
If y(x) ∈ Fp[[x]] is algebraic then y satisfies an equation of the shape
sX
i=r
fi(x)y pi
= 0 ,
with r , s ∈ N , the fi ∈ Fp[x] and fr = 0 . In fact, we may take r = 0 ,
for if r > 0 then writing fi = P
α x αf p
we get that
iα
Hence Ps−1 pi
i=r−1
fi+1,α(x)y
X
x α` s X
fiα(x)y pi−1´ p
= 0 .
α
i=r
= 0 and some frα = 0 .
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